As for any restricted Lie algebra, \(H^1_*\big (\mathfrak {m}_2 ^\lambda (p)\big )\)  consists of those ordinary cohomology classes \([\psi ]\in H^1 \big (\mathfrak {m}_2^\lambda (p)\big )\)  for which \(\operatorname{ind}^1(\psi )=0\)  (see [1]} or [2]}). If \(\psi =\sum _{k=1}^p \mu _ke^k\)  is any ordinary cocycle, then \(\mu _p=0\)  (\(p\ge 5\) ) so that for any
\(g\in \mathfrak {m}_2^\lambda (p)\) , we have
\(\operatorname{ind}^1(\psi )(g)=\psi \big (g^{[p]}\big )=\mu _p\left(\sum _{k=1}^p \alpha _k^p\lambda _k\right)=0,\) 
